UFCalcSet18 - x-coordinates at which the function g x = x...

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Exercises UF Calculus Set 18 1. Find the derivative of the function. State the domain of the derivative. (a) f ( x ) = ln(1 + x 2 ) (b) f ( x ) = [ ln( x ) ] 3 (c) f ( x ) = sin(ln( x )) (d) f ( x ) = x ln(sec( x )) (e) f ( x ) = ln(5 + e ) (f) f ( x ) = p 1 + ln( x ) (g) f ( x ) = ln ± 1 + 1 x ² (h) f ( x ) = 4 - ln( x ) 2 + ln( x ) (i) f ( x ) = log 3 (1 + x ) (j) f ( x ) = ln | x - e x | 2. Write the first derivative of the given functions. For which x -values do they have horizontal tangents? Assume α and β are positive constants. (a) f ( x ) = x 5 / 2 ln( x ) (b) f ( x ) = ln( x ) x 5 / 2 (c) f ( x ) = x 2 [ ln( x ) ] 2 (d) f ( x ) = [ ln( x ) ] 2 x 2 (e) f ( x ) = x α [ ln( x ) ] β (f) f ( x ) = [ ln( x ) ] β x α 3. For each curve below, write the equation of the tangent line at the given point: (a) y = x - ln( x ) at the point ( e,e - 1 ) . (b) y = ln(ln( x )) at the point having x -coordinate e . 4. Use logarithmic differentiation to calculate the derivative. (a) f ( x ) = e 2 x 1 + x 2 tan 2 ( x ) (b) f ( x ) = ± 1 + 2 x 2 4 - 5 x ² 4 (c) f ( x ) = x x (d) f ( x ) = (1 + sin( x )) 1 /x 5. By two different methods, find the derivative and all nonzero
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Unformatted text preview: x-coordinates at which the function g ( x ) = x α e βx has horizontal tangent lines. Assume α and β are positive constants. (a) using product and chain rules (b) using logarithmic differentiation 6. Find the derivative of the function y defined implicity in terms of x . (a) y 2 = ln(2 x + 3 y ) (b) ln(cos( y )) = 2 x + 5 (c) (1 + x ) y = y + x 7. Find, as directed, the equation of the tangent line to the curve implicitly defined as a function of x . (a) y + e y = 1 + ln( x ) at the point ( 1 , 0 ) (b) y 2-e 2 x = e x ln( y ) at the point ( 0 , 1 ) ——————————————————————-...
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This note was uploaded on 05/17/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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UFCalcSet18 - x-coordinates at which the function g x = x...

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